We find new bounds on the conformal dimension of small cancellation groups.
These are used to show that a random few relator group has conformal dimension
2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of
relators grows like l^K in the length l of the relators, then a.a.s. such a
random group has conformal dimension 2+K+o(1). In Gromov's density model, a
random group at density d<1/8 a.a.s. has conformal dimension ≍dl/∣logd∣.
The upper bound for C'(1/8) groups has two main ingredients:
ℓp-cohomology (following Bourdon-Kleiner), and walls in the Cayley
complex (building on Wise and Ollivier-Wise). To find lower bounds we refine
the methods of [Mackay, 2012] to create larger `round trees' in the Cayley
complex of such groups.
As a corollary, in the density model at d<1/8, the density d is determined,
up to a power, by the conformal dimension of the boundary and the Euler
characteristic of the group.Comment: v1: 42 pages, 21 figures; v2: 44 pages, 20 figures. Improved
exposition, final versio